Corners and simpliciality in oriented matroids and partial cubes

TL;DR

This paper advances the understanding of oriented matroids and partial cubes by proving new structural properties, refuting a longstanding conjecture, and introducing corners of complexes of oriented matroids, with implications for several open problems.

Contribution

It generalizes classical problems in oriented matroids and partial cubes, introduces the concept of corners in COMs, and confirms conjectures for specific classes of these structures.

Findings

Every element of Mandel's class of OMs is incident to a simplicial tope.

Mutation graphs of uniform OMs of order ≤ 9 are connected.

Realizable COMs and hypercellular graphs admit corner peelings.

Abstract

Building on a recent characterization of tope graphs of Complexes of Oriented Matroids (COMs), we tackle and generalize several classical problems in Oriented Matroids (OMs), Lopsided Sets (aka ample set systems), and partial cubes via Metric Graph Theory. Our first main result is that every element of an OM from a class introduced by Mandel is incident to a simplicial tope, i.e, such OMs contain no mutation-free elements. This allows us to refute a conjecture of Mandel from 1983, that would have implied the famous Las Vergnas' simplex conjecture. Further, we show that the mutation graph of uniform OMs of order at most 9 are connected, thus confirming a stronger conjecture of Cordovil-Las Vergnas in this setting. The second main contribution is the introduction of corners of COMs as a natural generalization of corners in Lopsided Sets. Generalizing results of Bandelt and Chepoi, Tracy Hall, and Chalopin et al. we prove that realizable COMs, rank 2 COMs, as well as hypercellular graphs admit corner peelings. Using this, we confirm Las Vergnas' conjecture for antipodal partial cubes (a class much lager than OMs) of small rank or isometric dimension.